Convex duality and the Skorokhod Problem. I

Probability Theory and Related Fields

Volumn 115, Issue 2, 1999, Pages 153-195

  Dupuis, Paul ^a   Ramanan, Kavita ^b  

^a BROWN UNIVERSITY   (United States)

^b LUCENT TECHNOLOGIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18544395348     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004400050269     Document Type: Article

References (44)

1. R. Anderson and S. Orey. Small random perturbations of dynamical systems with reflecting boundary. Nagoya Math. J., 60:189-216, 1976.
2. F. Avram, D. Bertsimas and M. Ricard. Optimization of multiclass queueing networks: a linear control approach. Stochastic Networks, IMA Volumes in Mathematics and Its Applications, F.P. Kelly and R.J. Williams, editors, 199-234, 1995.
3. A. Bernard and A. El Kharroubi. Régulation de processus dans le premier orthant de IR. Stochastics and Stochastics Reports, 34:149-167, 1991.
4. A. Buddhiraja and P. Dupuis. Necessary and Sufficient Conditions for the Stability of Constrained Processes. To appear in SIAM J. on Appl. Math.
5. H. Chen and A. Mandelbaum. Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. of Prob., 19:1436-1519, 1991.
6. H. Chen and H. Zhang. Diffusion approximations for multiclass FIFO queueing networks. Technical Report, 1995.
7. C. Costantini. The Skorokhod oblique reflection problem and a diffusion approximation for a class of transport processes. PhD thesis, University of Wisconsin, Madison, 1987.
8. J. Dai. Steady state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications. PhD thesis, Operations Research Dept., Stanford University, 1990.
9. J. G. Dai and S. P. Meyn. Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Transactions on Automatic Control, 40:1889-1904, 1995.
10. P. Dupuis and H. Ishii. SDE's with oblique reflection on nonsmooth domains. Ann. of Prob., 21:554-580, 1993.
11. P. Dupuis and H. Ishii. On oblique derivative problems for fully nonlinear second-order elliptic PDE's on domains with corners. Hokkaido Math. J., 20:135-164, 1991.
12. P. Dupuis and H. Ishii. Invariant sets for collections of oblique projections. Technical report, University of Massachusetts, 1991.
13. P. Dupuis and H. Ishii. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics, 35:31-62, 1991.
14. P. Dupuis and H. J. Kushner. Stochastic approximation and large deviations: upper bounds and w.p.1 convergence. SIAM J. on Control and Optimization, 27:1108-1135, 1989.
15. P. Dupuis and A. Nagurney. Dynamical systems and variational inequalities. Annals of Op. Res., 44:9-42, 1993.
16. P. Dupuis and K. Ramanan. A Skorokhod problem formulation and large deviation analysis of a processor sharing model. Queueing Systems, 28:109-124, 1998.
17. P. Dupuis and K. Ramanan. Convex duality and the Skorokhod Problem - II. To appear in Prob. Theor. and Rel. Fields, 1999.
18. P. Dupuis and K. Ramanan. Convex duality and the Skorokhod Problem. LCDS Technical Report 97-1, Brown University, 1997.
19. P. Dupuis and R. J. Williams. Lyapunov functions for semimartingale reflection Brownian motions. Ann. of Prob., 22:680-702, 1994.
20. A. N. Fillipov. Differential equations with discontinuous right-hand side. Mat. Sbornik (N.S), 51:99-128, 1960.
21. J. M. Harrison. Brownian Motion and Stochastic Flow Systems. Wiley, New York, 1985.
22. J. M. Harrison and M.I. Reiman. Reflected Brownian motion on an orthant. Ann. of Prob., 9:302-308, 1981.
23. L. Horvath. Strong approximations of open queueing networks. Math. of Oper. Res., 17:487-508, 1992.
24. F. Kelly and R. J. Williams, editors. Stochastic Networks, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New York, 1995.
25. F. Kelly, S. Zachary and I. Ziedins, editors. Stochastic Networks: Theory and Applications, Oxford University Press, London, 1996.
26. F. P. Kelly and C. N. Laws. Dynamic routing in open queueing networks : Brownian models, cut constraints and resource pooling. Queueing Systems, 13:47-86, 1993.
27. V. S. Kozyakin, A. Mandelbaum, and A. A. Vladimirov. Absolute stability and dynamic complementarity. Technical report, Technion, 1993.
28. H. J. Kushner. Control of trunk line systems in heavy traffic. SIAM J. on Control and Opt., 33:765-803, 1995.
29. P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Comm. Pure and Appl. Math., 37:511-553, 1984.
30. A. Mandelbaum and L. Van der Heyden. Complementarity and reflection. Unpublished work, 1987.
31. L. F. Martins and H. J. Kushner. Routing and singular control for queueing networks in heavy traffic. SIAM J. on Control and Optimization, 28:1209-1233, 1990.
32. V. Nguyen. Processing networks with parallel and sequential tasks: heavy traffic analysis and Brownian limits. Ann. of App. Prob., 3:28-55, 1993.
33. G. Pats. State Dependent Queueing Networks: Approximation and Applications. PhD thesis, Department of Industrial Engineering and Management, Technion, Haifa, Israel, 1996.
34. W. P. Peterson. Diffusion approximations for networks of queues with multiple customer types. Math. of Oper. Res., 9:90-118, 1991.
35. M. I. Reiman and R. J. Williams. A boundary property of semimartingale reflecting Brownian motions. Prob. Theor. Rel. Fields, 77:87-97, 1988.
36. M. I. Reiman. Open queueing networks in heavy traffic. Math. of Oper. Res., 9:441-458, 1984.
37. H. L. Royden. Real Analysis. Macmillan Publishing Company, New York, 1989.
38. Y. Saisho. Stochastic differential equations for multi-dimensional domain with reflecting boundary. Prob. Theor. Rel. Fields, 74:455-477, 1987.
39. A. V. Skorokhod. Stochastic equations for diffusions in a bounded region. Theor. of Prob. and Its Appl., 6:264-274, 1961.
40. H. Tanaka. Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J., 9:163-177, 1979.
41. P. Tsoucas. Rare events in series of queues. J. of App. Prob., 29:168-175, 1992.
42. L. M. Wein. Optimal control of a two station Brownian network. Math. of Oper. Res., 15:215-242, 1990.
43. R. J. Williams. Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. of Prob., 13:758-778, 1985.
44. R. J. Williams and S. R. S. Varadhan. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math., 24:147-225, 1985.